Optimal. Leaf size=62 \[ -\frac{\text{Chi}(b x) \sinh (b x)}{b^2}+\frac{\text{Shi}(2 b x)}{2 b^2}-\frac{\sinh (b x) \cosh (b x)}{2 b^2}+\frac{x \text{Chi}(b x) \cosh (b x)}{b}-\frac{x}{2 b} \]
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Rubi [A] time = 0.0715016, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {6549, 12, 2635, 8, 6541, 5448, 3298} \[ -\frac{\text{Chi}(b x) \sinh (b x)}{b^2}+\frac{\text{Shi}(2 b x)}{2 b^2}-\frac{\sinh (b x) \cosh (b x)}{2 b^2}+\frac{x \text{Chi}(b x) \cosh (b x)}{b}-\frac{x}{2 b} \]
Antiderivative was successfully verified.
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Rule 6549
Rule 12
Rule 2635
Rule 8
Rule 6541
Rule 5448
Rule 3298
Rubi steps
\begin{align*} \int x \text{Chi}(b x) \sinh (b x) \, dx &=\frac{x \cosh (b x) \text{Chi}(b x)}{b}-\frac{\int \cosh (b x) \text{Chi}(b x) \, dx}{b}-\int \frac{\cosh ^2(b x)}{b} \, dx\\ &=\frac{x \cosh (b x) \text{Chi}(b x)}{b}-\frac{\text{Chi}(b x) \sinh (b x)}{b^2}-\frac{\int \cosh ^2(b x) \, dx}{b}+\frac{\int \frac{\cosh (b x) \sinh (b x)}{b x} \, dx}{b}\\ &=\frac{x \cosh (b x) \text{Chi}(b x)}{b}-\frac{\cosh (b x) \sinh (b x)}{2 b^2}-\frac{\text{Chi}(b x) \sinh (b x)}{b^2}+\frac{\int \frac{\cosh (b x) \sinh (b x)}{x} \, dx}{b^2}-\frac{\int 1 \, dx}{2 b}\\ &=-\frac{x}{2 b}+\frac{x \cosh (b x) \text{Chi}(b x)}{b}-\frac{\cosh (b x) \sinh (b x)}{2 b^2}-\frac{\text{Chi}(b x) \sinh (b x)}{b^2}+\frac{\int \frac{\sinh (2 b x)}{2 x} \, dx}{b^2}\\ &=-\frac{x}{2 b}+\frac{x \cosh (b x) \text{Chi}(b x)}{b}-\frac{\cosh (b x) \sinh (b x)}{2 b^2}-\frac{\text{Chi}(b x) \sinh (b x)}{b^2}+\frac{\int \frac{\sinh (2 b x)}{x} \, dx}{2 b^2}\\ &=-\frac{x}{2 b}+\frac{x \cosh (b x) \text{Chi}(b x)}{b}-\frac{\cosh (b x) \sinh (b x)}{2 b^2}-\frac{\text{Chi}(b x) \sinh (b x)}{b^2}+\frac{\text{Shi}(2 b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0737158, size = 44, normalized size = 0.71 \[ -\frac{\text{Chi}(b x) (4 \sinh (b x)-4 b x \cosh (b x))-2 \text{Shi}(2 b x)+2 b x+\sinh (2 b x)}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 46, normalized size = 0.7 \begin{align*}{\frac{1}{{b}^{2}} \left ({\it Chi} \left ( bx \right ) \left ( bx\cosh \left ( bx \right ) -\sinh \left ( bx \right ) \right ) -{\frac{\cosh \left ( bx \right ) \sinh \left ( bx \right ) }{2}}-{\frac{bx}{2}}+{\frac{{\it Shi} \left ( 2\,bx \right ) }{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Chi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \operatorname{Chi}\left (b x\right ) \sinh \left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sinh{\left (b x \right )} \operatorname{Chi}\left (b x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Chi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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